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Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

   
 
 

 

Asymptotic results for certain first-passage times and areas of renewal processes


Authors: Claudio Macci and Barbara Pacchiarotti
Journal: Theor. Probability and Math. Statist. 108 (2023), 127-148
MSC (2020): Primary 60F10, 60F05, 60K05
DOI: https://doi.org/10.1090/tpms/1189
Published electronically: May 2, 2023
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Abstract: We consider the process $\{x-N(t):t\geq 0\}$, where $x\in \mathbb {R}_+$ and $\{N(t):t\geq 0\}$ is a renewal process with light-tailed distributed holding times. We are interested in the joint distribution of $(\tau (x),A(x))$ where $\tau (x)$ is the first-passage time of $\{x-N(t):t\geq 0\}$ to reach zero or a negative value, and $A(x)≔\int _0^{\tau (x)}(x-N(t))dt$ is the corresponding first-passage (positive) area swept out by the process $\{x-N(t):t\geq 0\}$. We remark that we can define the sequence $\{(\tau (n),A(n)):n\geq 1\}$ by referring to the concept of integrated random walk. Our aim is to prove asymptotic results as $x\to \infty$ in the fashion of large (and moderate) deviations.


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Additional Information

Claudio Macci
Affiliation: Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Rome, Italy
Email: macci@mat.uniroma2.it

Barbara Pacchiarotti
Affiliation: Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Rome, Italy
Email: pacchiar@mat.uniroma2.it

Keywords: Large deviations, moderate deviations, joint distribution, integrated random walk
Received by editor(s): October 2, 2021
Accepted for publication: April 5, 2022
Published electronically: May 2, 2023
Additional Notes: The authors acknowledge the support of GNAMPA-INdAM and of MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata (CUP E83C18000100006).
Article copyright: © Copyright 2023 Taras Shevchenko National University of Kyiv